Brauer Type Embedding Problems

Galois realizability of groups of orders p 5 and p 6

“…Ledet describes in his book [9] a more general formula for the decomposition of the obstruction of µ -embedding problems with finite group F isomorphic to a direct product of two groups. Let G be arbitrary finite group, and be a prime divisor of ord G. Define O (G) as the subgroup of G generated by all elements of order prime to .…”

Induced orthogonal representations of Galois groups

“…We adopt the notations about Clifford algebras used in [Le,Ch. 5,§2]: C (q) is the Clifford algebra of q; C 0 (q) is the even Clifford algebra;…”

Equivariant vector fields on non-trivial SOn-torsors and differential Galois theory

“…An excellent reference for quadratic forms is [5]. For a more comprehensive review of the basic facts than the one provided here, see also Chapter 7 in [6].…”

“…It is well known that the Galois cohomology with coefficients in the special orthogonal group H 1 (K, SO n ) is in bijective correspondence with the isomorphism classes of regular n-dimensional quadratic forms over K of discriminant 1, under Galois twist [6,Theorem 5.3.1]. It is also the case that for any linear algebraic group G defined over K, the cohomology classes in H 1 (K, G) correspond to isomorphism classes of G-torsors [9,Proposition 33].…”